Schur m-Power Convexity of a Class of Multiplicatively Convex Functions and Applications

Abstract

We investigate the conditions under which the symmetric functions are Schur $m$-power convex for $x\in {\mathbb{R}}_{++}^n$ and $r>0$. As a consequence, we prove that these functions are Schur geometrically convex and Schur harmonically convex, which generalizes some known results. By applying the theory of majorization, several inequalities involving the $p$th power mean and the arithmetic, the geometric, or the harmonic means are presented.